Can all composite numbers be factored into a finite set of prime numbers? For example, the set (2,3,5,7,11) can make up the the factors of an extraordinary amount of large numbers. Is there a maximum number of primes that exist to make up the product of all composite numbers?
2026-03-29 15:40:29.1774798829
Can all composite numbers be factored into a finite set of prime numbers?
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Every such maximal set $S$ must coincide with the set of primes $\Bbb P$. To show this, suppose that your maximal set is missing a prime $p$ and consider the composite number $2p$. Because of the Fundamental Theorem of Arithmetic, we cannot factor $2p$ with different primes. This is a contradiction and we are done.